Download Force and Motion I 4.0

Team ________________
________________
Force and Motion
In previous labs, you used a motion detector to measure the position, velocity, and acceleration
of moving objects. You were not concerned about the “mechanism” that caused the object to
move, i.e. the forces that acted on the object. In other words, you answered the kinematic
question “How do objects move?” Now it is time to answer the dynamic question “Why do
objects move?” From your experiences, you know that force and motion are related. To open a
door, you must push or pull on the handle. To stop a car, you must step on the brakes. The force
of gravity causes the earth to move around the sun. The force of electricity causes the electron in
an atom to move around the proton.
Newton’s celebrated equation of motion, F = ma , describes the precise connection between
force (F) and motion (a). You have used a motion sensor (accelerometer) to measure a. You
will now use a “force-ometer” to measure F. By measuring the left side of F = ma with a force
meter and the right side with a motion meter, you will discover the deep relation between F and
ma firsthand in the laboratory.
Thought Experiment
A 2 kg mass and a 3 kg mass are connected by a massless cord and move on a horizontal
frictionless surface. The 3 kg mass is pulled to the right with a force of 20 N. The tension in the
cord is 8 N.
2 kg
8N
20 N
3 kg
Force:
Net Force on 2 kg mass
=
__________ N .
Net Force on 3 kg mass
=
_____
−
_____
=
__________
N.
Motion:
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Mass x Acceleration of 2 kg mass =
_____ x
_____
=
__________
kg m/s .
Mass x Acceleration of 3 kg mass =
_____ x
_____
=
__________
kg m/s .
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Part I. “Forceometers”
The Spring Scale
The spring scale is the prototype “forceometer”. For centuries, springs have been used to
measure force. The force required to stretch a spring is proportional to the amount of stretch. A
spring scale is a spring that has been calibrated to convert the amount of stretch (meters) into the
corresponding value of the force (Newtons).
Examine the spring scale. Note that there is a large fixed hook at one end and a small hook at the
other end. The small hook is attached to a sliding rod, which is connected to the spring. Pull on
the small hook so that the spring is stretched several centimeters (half its maximum stretch) and
the reading on the force scale is 0.5 N. Double the stretch of the spring so that the force is 1.0 N
(maximum scale reading).
You already know what a “one pound force” feels like. But what does a “one Newton force” feel
like? Pull on the spring scale until you have a good kinesthetic sense of a one Newton force.
Zero the spring scale as follows. Hold the spring scale in the vertical position with the small
hook downward. With nothing attached to the hook, make sure the scale reads zero. If the
reading is not “0” , then try another spring scale or consult your instructor. Once the scale is
zeroed, hang a 50 gram mass from the small hook. What is the value of the force F recorded by
the spring scale?
Spring Scale Reading:
F =
______________ N.
Since the 50 gram mass is at rest, the upward pull of the spring on the mass is equal to the
downward pull of the earth on the mass. The force of gravity on an object is called the weight of
the object. Calculate the theoretical value of the weight of a 50 gram mass.
Calculated Weight of 50 gram mass:
W = ______________ N.
% diff between Measured Weight (spring scale reading) and Calculated Weight is ______ %.
The Force Sensor
Examine the force sensor. The switch on the force sensor should be set to the 10 N range. The
force sensor is an electronic strain gauge. Any force (push or pull) on the hook causes a metal
block inside the sensor to bend slightly. The strain in the block is converted into an electrical
signal (voltage) which is then converted, via the interface box (ULI), into the corresponding
value of the mechanical force. So like the spring scale, the force sensor converts a distance
(strain in meters) into a force (Newtons).
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The force sensor measures F(t) = Force on the hook as a function of time. To activate the force
sensor or “force probe”, start Logger Pro and open the file ForceProbe. Click on Collect. Push
and pull on the hook and note the digital values of F(t) that are displayed in the data table and the
F(t) curve that is traced out in the graph window.
Zero the force sensor as follows. While holding the force sensor in the vertical position with the
hook downward and nothing attached to the hook, click on the Zero button (to the right of the
Collect button). Once the sensor is zeroed, hang a 50 gram mass from the hook and click on
Collect. Average the F(t) data. What is the value of the force F recorded by the force sensor ?
Force Sensor Reading: F =
% diff between
______________ N.
Measured Weight (force sensor reading) and Calculated Weight is ______ %.
Part II. Apply F. Measure ma. Prove the Law F = ma .
Measure F with the Spring Scale
First make sure that the track is level using the leveler. The spring scale will be used to apply the
force F. Make sure that the spring scale reads “zero” when it is held in the horizontal position
and nothing is attached to the small hook.
Attach the small hook of the spring scale to the cart. Pull the cart along the track with a constant
force of 0.10 N. It may be a bit tricky to keep the force constant. In experimental physics,
nothing is exactly constant. It is natural for the force to fluctuate around 0.10 N. This error in
keeping the force constant should have little affect on your results. Sometimes you pull too hard,
sometimes you pull too soft, but on average you pull just right!
Practice pulling the cart until you get a “feel” for how to keep the force scale reading at 0.10 N.
While you are pulling, carefully note the amount by which your applied force F varies above and
below 0.10 N. This up and down variation in F determines the uncertainty in F. For example, if
the spring scale reading varies between 0.50 N and 0.15 N during the pull, then you would report
the experimental value of the force as F = 0.10 N ± 0.05 N. If your pull is steadier, then the
uncertainty in the force will be smaller.
Based on your pulling trials and observed range of force values, report the measured value of
your applied force:
F =
0.10 N
±
________ N .
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Measure ma with a Stopwatch
An accurate way to determine the constant acceleration a of your cart is to measure time and
space. Use a stopwatch to measure the time t it takes for your cart − starting from rest and
pulled with the force 0.10 N − to cover the distance d = 1.0 m along the track. Use the kinematic
relation d = ½ at2 to compute a from your measured values of t and d. Repeat the run five times.
Record your five measured values of t in the motion table below. For each run, compute the
values of a and ma .
Measure the mass m of your cart using the mass scale.
Motion Table
d (m)
1.0
1.0
1.0
1.0
1.0
t (s)
a (m/s2)
m = _____________ kg .
ma (N)
Compute the average value of ma. Estimate the uncertainty from the spread in your five values
of ma : Uncertainty = (mamax − mamin) / 2.
ma = _____________
±
____________ N .
Compare ma and F
You have measured the force quantity F using a spring scale. In a completely independent
measurement, you have measured the motion quantity ma using a stopwatch (and meter stick and
mass scale). The big question is this: Is F equal to ma ?
Remember, since an experimental value (number ± uncertainty) is really a range of numbers,
“comparing two values” really means “comparing two ranges”. Display your results for F and
ma on the following range diagram. Remember that an experimental value, such as 0.10 ± 0.03
N, is displayed on the range diagram as a horizontal line segment from 0.07 N to 0.13 N. Write
the numerical values on the axis labeled “Newtons”.
F
ma
Newtons
Use the results displayed on your experimental range diagram to rigorously answer the deep
question: Is F equal to ma ?
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Double the Force
Use the spring scale to pull the cart with the contant force F = 0.20 N . Use the stopwatch to
measure the time t it takes for the cart, starting from rest, to traverse the distance d = 1.0 m along
the track. Repeat the run one more time. Average your two values of t. Compute the value of a.
a = ___________ m/s2.
t = ___________ s .
When you doubled the force on the cart (F → 2F), what happened to the acceleration of the cart?
% difference between
F =
0.20 N_
and
ma = _
N
is
_____ % .
Double the Mass
Place the long bar weight on top of your cart.
Measure the mass of the loaded cart.
m = ____________ kg .
Pull the cart with the constant force F = 0.10 N . Measure the time t it takes for the cart,
starting from rest, to cover the distance d = 1.0 m along the track. Repeat the run. Average your
two values of t. Compute the value of a.
a = ___________ m/s2.
t = ___________ s .
When you doubled the mass of the cart (m → 2m and applied the same 0.10 N force), what
happened to the acceleration of the cart?
% difference between
F =
0.10 N_
and
ma = _
N
is
_____ % .
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Part III. Two Body Dynamics. Net Force.
In the previous part, you studied a system consisting of one mass (body). Here you will study a
mechanical system composed of two connected masses. In physics, the “two-body problem”
refers to any dynamical problem involving two interacting objects. Solving the equations of
motion for the sun-earth-moon system is a famous “three-body problem” in theoretical physics.
You will also discover how the net force − the sum of all the individual forces − acting on a body
is the crucial dynamical quantity that determines the motion of the body. The all-encompassing
goal of classical mechanics: FIND THE NET FORCE !
A. Two-Cart “Train”
Build the following two-body mechanical system. Body 1 consists of cart 1 plus spring scale 1.
Body 2 consists of cart 2 plus spring scale 2. Make sure the spring scales are Zeroed. Both
scales should read “0” when held in the horizontal position and nothing is attached to the small
hook. Note the orientation of the scales: large hook on left, small hook on right. Use one or two
rubber bands to secure the scale to the cart.
scale 1
scale 2
Cart 1
Cart 2
The experiment consists of pulling the system to the right with a contant force. Here is the
Newtonian schematic − the textbook picture − of this real-world mechanical system:
m1
F12
m2
Fpull
System parameters:
m1
m2
F12
Fpull
=
=
=
=
mass of cart 1 plus mass of spring scale 1.
mass of cart 2 plus mass of spring scale 2.
Connecting Force (Force on 1 due to 2), recorded by spring scale 1.
Pulling Force, recorded by spring scale 2.
Note that the Net Force on m2 is Fpull − F12 .
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Measure Mass
Use the mass scale to measure the mass of body 1 and the mass of body 2. Remember that each
body consists of one cart plus one spring scale.
m1 =
____________ kg .
m2 =
____________ kg .
Measure Force
The connecting force F12 is the force of interaction between body 1 and body 2. It is the force
exerted on body 1 due to body 2. By Newton’s Third Law, it is equal and opposite to the force
on body 2 due to body 1: F12 = − F21 . Examine this force of interaction in your actual two-cart
system. Note how the sliding rod of spring scale 1 provides the physical connection between the
carts. The stretched spring between cart 1 and cart 2 determines the value of F12 . The bigger the
stretch, the larger the force.
Place your two-cart system directly on the table surface, not on the metal track. This will
minimize friction and allow you to pull the carts over a longer distance. Hold the small hook of
spring scale 2 and pull so that the scale always reads “ 0.2 N ”. While you are pulling, your
team member will constantly monitor the reading on scale 1, thereby measuring the force of
interaction F12 . Repeat this pulling/monitoring until you are confident (to within 10%) in your
measured value of force. Report your value of F12 :
Fpull =
0.20
N.
F12
=
___________ N .
Measure Acceleration
Find the acceleration a of the two-cart system as you pull with the constant force Fpull = 0.2 N.
Use the stopwatch to measure the time t it takes the system, starting from rest, to cover a distance
d = 1.0 m along the table surface. Lay a meter stick (x axis) on the table surface and let the twocart system move alongside the stick from one end to the other end. Repeat one or two times and
average your values of t. Compute the value of a from your measured value of t (and d = 1.0 m).
t = __________ s .
a = ____________ m/s2.
Experimental Proof of Fnet = ma
Draw the free body diagrams for m1 and m2 in the space below. Only include the forces that
point in the horizontal direction (the vertical motion is trivial). Write the numerical value of the
force next to each force vector. Draw the length of each vector to scale (approximately),
i.e. if v = 2 and u = 6, then draw the length of vector u to be three times the length of vector v.
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Free Body 1
Free Body 2
Sum the forces to find the net force Fnet on each body. Multiply the mass and the acceleration to
find the motion quantity ma for each body.
Fnet 1 =
m1 a1 =
___________ N .
Fnet 2 =
___________ kg m/s2 .
m2 a2
=
___________ N .
___________ kg m/s2 .
Conclusion
Do your measured values of Fnet and ma for each body provide an experimental proof of
Newton’s Second Law of Motion to within a 10 % margin of experimental error? Explain.
Changing the Mass. Testing the Law Fnet = ma .
Add the long bar weight to the top of cart 1. Measure the mass of body 1 (cart + scale + weight).
m1 =
____________ kg .
Pull the two-body system with a constant force of 0.30 N over a distance of d = 1.0 m. Measure
the values of F12 and a as before by monitoring scale 1 and using a stopwatch.
Fpull =
0.30
N.
F12 = ___________ N .
t = __________ s.
a =
___________ m/s2 .
Find the net force on each body and the mass-times-acceleration of each body.
Fnet 1 = __________ N .
m1 a1 =
______ x ______ =
___________ kg m/s2 .
Fnet 2 = __________ N .
m2 a2 =
______ x ______ =
___________ kg m/s2 .
Are your experimental results consistent with Newton’s Second Law of Motion? Explain.
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Remove the bar weight from cart 1 and add it to the top of cart 2. Measure the mass of body 2
(cart + scale + weight).
m2 =
____________ kg .
Pull the two-body system with a constant force of 0.3 N over a distance of d = 1.0 m. Measure
the values of F12 and a as before by monitoring scale 1 and using a stopwatch.
Fpull =
0.30
N.
F12 = ___________ N .
t = __________ s.
a =
___________ m/s2 .
Find the net force on each body and the mass-times-acceleration of each body.
Fnet 1 = __________ N .
m1 a1 =
______ x ______ =
___________ kg m/s2 .
Fnet 2 = __________ N .
m2 a2 =
______ x ______ =
___________ kg m/s2 .
Are your experimental results consistent with Newton’s Second Law of Motion? Explain.
Summary Exercise
The following three pictures summarize your three experiments, but with simple whole numbers
for the forces and masses that make the physics (Fnet = ma) transparent and the calculations easy.
The pictures show two masses connected by a massless cord. The system is pulled with a force
of 6 N on a horizontal frictionless surface. Based on pure theory, find the values of the tension
(N) in the cord and the acceleration (m/s2) of the system. Write your results directly on the
pictures.
1 kg
N
1 kg
6N
m/s2
2 kg
N
1 kg
6N
m/s2
1 kg
N
2 kg
6N
m/s2
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B. Rolling Cart Connected to Falling Weight
Build the following two-body mechanical system. Body 1 is the cart plus the force sensor. Body
2 is the hanging weight.
force
sensor
motion
sensor
cart
string
pulley
track
200 gram
weight
Measure m , a , F
In this experiment, you will use the mass scale to measure m , the motion sensor to measure a ,
and the force sensor to measure F. The mass of body 2 is given to be 200 grams. Measure the
mass of body 1 (cart + force probe).
m1 =
____________ kg .
m2 =
0.200
kg .
Open the file ForceVelAcc. With nothing attached to the hook on the force sensor (remove the
string), zero the force sensor by clicking on the Zero button. Attach the string to the force sensor
hook. Hold the cart with your hand so that the system is at rest. First click on Collect. Wait
until you hear the sonic clicks from the motion sensor. Then release the cart.
The motion sensor will record the motion functions v(t) and a(t). The force sensor will record
the force function F(t). Note: F(t) = Tension T in the string as a function of time.
Examine the force and motion graphs. You may have to change the scales in order to magnify
the good data region. Since gravity and tension are constant forces, the F(t) and a(t) graphs
should be horizontal (flat) lines, with slight up and down fluctuations (due to experimental
errors) around the average value. To find the tension force T in your experiment, you can
average the good F(t) data. To find the acceleration a , you can compute the slope of the good
v(t) line or you can average the good a(t) data.
Tension T = ____________ N .
Acceleration a =
____________ m/s2 .
Before you release the cart (system is at rest), the tension in the string is
___________ N.
Experimental Proof of Fnet = ma
Draw the free body diagrams for m1 and m2 in the space below. Write the numerical value of
the force next to each force vector. Draw the length of each vector to scale (approximately).
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Free Body 1
Free Body 2
Sum the forces to find the net force Fnet on each body.
Fnet 1 =
___________ N .
Fnet 2 =
___________ N .
Multiply the mass and the acceleration to find the motion quantity ma for each body.
m1 a1 =
________ x ________
m2 a2 =
_____________ kg m/s2 .
=
=
________ x ________
_____________ kg m/s2 .
Conclusion
Do your measured values of Fnet and ma for each body provide an experimental proof of
Newton’s Second Law of Motion to within a 10 % margin of experimental error? Explain.
Thought Experiment
Suppose the masses of the two bodies in your experiment were m1 = 2 kg and m2 = 5 kg. Do not
perform this experiment. Fill in the “F” and “ma” blanks below.
2 kg
Fnet 1 =
_______
m1 a1 =
2 x ___ =
_______
5 kg
Fnet 2 = ___ − ___
=
_______
m2 a2 =
=
_______
5 x ___
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Part IV. Creating a Physics Sculpture
Title:
“Motionless Bodies on a Tilted Beam.
Gravity and Tension in Perfect Harmony”
Cart
Weight (150 gram)
Track
Box
Depending on the tilt of the track, the cart will accelerate up or down. However, there exists one
special tilt angle such that the two-mass system is in equilibrium − a special state where the
masses do not move at all! In this state of zero acceleration , the net force on each mass is
equal to zero. In the sculpture above, the forces of gravity and tension are in perfect balance.
Design Goal: Where should you place the box so that the masses do not move? Work out the
theory behind this mechanical structure before you actually build the structure.
The Theory
Here is a schematic of the system showing the system parameters m1 , m2 , L , H .
m1
m2
L
H
θ
Note that the variable L specifies the location of the box of height H. More specifically, the
upper left corner of the box touches the track at a point that is a distance L away from the point
where the track touches the table. Note the “tilt relation” sin θ = H/L . Answer the following
questions using only symbols. No numbers allowed.
1. Draw two free body diagrams, one for m1 and one for m2 .
m1
m2
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2. Set up Newton’s equation of motion Fnet 1 = m1 a for body 1 and Newton’s equation of
motion Fnet 2 = m2 a for body 2.
3. Simplify the motion equations using the motionless (equilibrium) condition a = 0 and the tilt
relation sin θ = H/L .
4. Solve Newton’s equations for L as a function of m1 , m2 , and H.
L =
Have your instructor check your theoretical formula for L before your build your sculpture.
Measure the values of H and m1 . The value of m2 is given to be 150 grams.
H = _________ m .
m1 = _________ kg .
m2 =
0.150
kg .
Plug the values of these system parameters into your theoretical formula for L.
L (theory)
=
_______________
m.
Building the Sculpture
First place the box under the far right end of the track (near the pulley) to achieve the smallest
possible tilt. Now increase the tilt by gently sliding the box along the table surface to the left
(while lifting the track slightly). Stop and check for equilibrium − release the cart (perhaps give
it a nudge) and observe the motion. Continue to increase the tilt until you find the unique
location of the box for which the bodies remain motionless − the special state where gravity and
tension are in “perfect harmony”.
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In your search for this special state of equilibrium, you will find a small range of L values −
slightly greater than and slightly less than the “perfect” value − for which the system remains
motionless. Welcome to the real world of experimental errors and uncertainty!
What ingredient(s) did you leave out of your theoretical analysis that could account for this
experimental range of L values?
Report your experimental value of L (including the uncertainty) for which the system is in
equilibrium.
L (experiment)
=
____________
±
____________
m.
Compare Theory and Experiment
Does your value of L (theory) fall within the range of values defined by L (experiment)?
% difference between L (theory) and L (experiment) is
_______ % .
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